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Full text of "Treatise On Analysis Vol-Ii"

9 THE PLANCHEREL-GODEMENT THEOREM 377 To prove this, observe that for each G e 3fc(Sg) we have mg(GF) = mGF(l) and, by virtue of (15.9.2.10), WGF = G • WF; hence TWGF(!) = JG(%) dmF(x); but since mGF(l) = JG(x)F(%) dmg(x) by definition, this implies the relation (15.9.2.14). In particular, bearing in mind (15.9.2.12), we have (15.9.2.15) mx ,,=*]>• w, for all x, y in A. Since mx>y is a bounded measure, this relation implies that jcj) is mg'integrable (13.14.4). The functions x therefore all belong to , mg) ; also, it follows from (15.9.2.15) and (15.9.2.5) that, for all z e A, = f (15.9.2.16) g(zx9 y) = ffo) dmx,y(X) = In other words, we have proved (15.9.2.1) in the particular case where x is replaced by & product zx. It remains to show that (15.9.2.1) is true in general. The remark at the beginning of (15.9.2.3) shows that, for all operators KG.*;, (15.9.2.17) (V - ng(x) 1 7i,GO) = f JS'g in particular, taking V = lHg, we have (15.9.2.18) 9(x,y) But, from (15.9.2.15), we have f jsg Since the measure mXty is by definition induced by fj,Xty on Sg , it follows that what we have to prove is that, when 0 e S£ (and therefore Sg = S^ — {0}) (15.9.2.19) ^({0}) = 0. Now it follows immediately from the definition of ^x>y that the mapping (x, y) i— >• {j,Xt y is sesquilinear, and hence (13.16.1) the complex-valued func- tion (x, jO^/^./W) is a sesquilinear form on A x A. Moreover, it follows directly from (15.9.2.17) and the Gelfand-Neumark theorem (15.4.14) that, for each continuous function F on the compact space S^ = S^ u {0}, IK' Uor each function F e <J>, we have